We will call a mixed metric any metric defined in a plane domain which changes its character - it is Remannian in one region and Lorentzian in the other. We investigate the existence of a normal form for our metric locally in a neighbourhood of a point belonging to the common boundary of these two regions. It is an analogue of the problem of existence of isothermal coordinates in the classical setting. We show that generically we have the unique conformal model $du^{2}+vdv^2}$. There are, however, some special points where the space of conformal models is of infinite dimension.